#382461

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3226 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 3226 conjugacy classes of subgroups — too many to draw. Only its 370 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) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,d^c=de

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268310, #72268317
maximal normal subgroup of	#72268310, #72268317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289657);