#832799

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
prime radical	C₂×#2537	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3084 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=fi,d^2=fi,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	#72297937, #72297939
maximal normal subgroup of	#72297937, #72297939
maximal quotient of	#72298533

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 739995);