#157142

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=g,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267425, #72267457, #72267462
maximal normal subgroup of	#72267425, #72267457, #72267462

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 64338);