#992933

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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

3454 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331631, #72331663, #72331692
maximal normal subgroup of	#72331631, #72331663, #72331692
maximal quotient of	#72332527

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 900129);