#394168

order	512
nilpotency class	3
isoclinism	isoclinic class of #394058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#34981	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283971, #72284018, #72284181, …
maximal normal subgroup of	#72283971, #72284018, #72284181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301364);