#147608

order	512
nilpotency class	4
isoclinism	isoclinic class of #129437
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

900 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6,10,15,11,16,7,17)(8,20,23,31,22,13,18,29)(9,14,19,27,25,30,24,12)(21,34,32,26,36,28,35,33);(1,2)(3,4)(5,7,11,10)(6,12,16,27)(8,18,22,23)(9,24,25,19)(13,26,20,33)(14,15,30,17)(21,35,36,32)(28,29,34,31);(2,4)(5,8,11,22)(6,13)(7,18,10,23)(9,21,25,36)(12,26)(14,28)(15,29)(16,20)(17,31)(19,32,24,35)(27,33)(30,34)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,g^b=ghi,e^c=ei,g^d=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, 54804);