#7054002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2323 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)(7,9)(8,10)(11,12)(13,14,18,22)(15,24,16,20)(17,23,19,21)(25,27,26,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,18)(17,26)(19,25)(21,28)(22,24)(23,27);(5,7,8,12)(6,9,10,11)(14,21)(15,16)(17,19)(20,27)(22,23)(24,28);(2,4)(7,12)(9,11)(13,16,18,15)(14,20,22,24)(17,25,19,26)(21,28,23,27);(6,10)(9,11)(13,17,18,19)(14,21,22,23)(15,25,16,26)(20,27,24,28)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309052, #72309057
maximal normal subgroup of	#72309052, #72309057

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961198);