#865888

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=ghi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331965, #72332008, #72332031
maximal normal subgroup of	#72331965, #72332008, #72332031
maximal quotient of	#72332566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 773084);