#764494

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

3582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332013, #72332026, #72332056
maximal normal subgroup of	#72332013, #72332026, #72332056
maximal quotient of	#72333156

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 671690);