#904625

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	#15278	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|c^2=hi,e^2=i,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	#72319866
maximal normal subgroup of	#72319866
maximal quotient of	#72320177, #72320198

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 811821);