#92990

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#99
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×#431	C₂²
socle	C₂²	#1280
prime radical	C₂⁴	#97
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×#431 ▷ C₂²×C₄ ▷ C₂² ▷ C₁

221 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=hi,g^2=i,b^a=bc,c^a=cf,f^a=fh,c^b=ci,d^b=dfi,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186);