#773117

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3892 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)(9,10)(11,15)(12,13)(14,16)(18,19)(22,23);(6,8)(10,13)(14,15)(17,18,21,19)(20,23,24,22);(5,6)(7,8)(17,18,21,19)(20,22,24,23);(1,2)(3,4)(9,11,12,16)(10,14,13,15)(17,19,21,18)(20,22,24,23);(9,10)(11,15)(12,13)(14,16)(17,20,21,24)(18,22,19,23)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261730, #72261745, #72261752
maximal normal subgroup of	#72261730, #72261745, #72261752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 680313);