#121953

order	512
nilpotency class	3
isoclinism	isoclinic class of #121952
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

361 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,11,7,10,5,21)(3,12,14,24,16,8,15,25)(4,17,20,9,19,27,22,28)(13,26,18,23,30,32,31,29)(33,34)(35,43)(36,40,48,56)(37,42)(38,41)(39,45)(44,53,62,61)(46,59)(47,55,63,64)(49,57,51,60)(50,54)(52,58);(1,3,5,14,7,16,6,15)(2,8,21,25,10,12,11,24)(4,18,22,30,19,31,20,13)(9,26,17,23,28,32,27,29)(33,35)(34,39,41,43)(36,47)(37,50)(38,45)(40,55,56,64)(42,59,58,54)(44,49)(46,52)(48,63)(51,62)(53,60,61,57);(1,4,5,22,7,19,6,20)(2,9,21,17,10,28,11,27)(3,13,14,18,16,30,15,31)(8,23,25,32,12,29,24,26)(33,36)(34,40,41,56)(35,44)(37,51)(38,48)(39,53,43,61)(42,60,58,57)(45,62)(46,63)(47,50)(49,52)(54,64,59,55)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=gh,d^c=dhi,e^c=ehi,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29149);