#843222

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

3400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,8)(9,10)(11,12)(15,19)(18,20)(21,22)(23,28)(24,26)(25,27);(6,10)(8,11)(13,14,17,16)(15,18,19,20)(21,23)(22,26)(24,25)(27,28);(1,2)(3,4)(21,22)(23,28)(24,26)(25,27);(5,7,9,12)(6,11,10,8)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,28)(25,27);(5,8,9,11)(6,7,10,12)(13,16,17,14)(15,20,19,18)(21,24,25,23)(22,26,27,28)

Polycyclic presentation:

2^9|b^2=fhi,d^2=fh,e^2=hi,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	#72278301
maximal normal subgroup of	#72278301
maximal quotient of	#72278768, #72279031, #72299085

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 750418);