#7053984

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

2433 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)(15,19)(16,21)(17,20)(18,22)(23,27)(24,28)(25,26);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,25)(17,24)(18,23)(20,28)(21,26)(22,27);(5,7,8,12)(6,9,10,11)(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(2,4)(7,12)(9,11)(13,15)(14,19)(16,25)(17,24)(18,23)(20,28)(21,26)(22,27);(6,10)(9,11)(13,16,18,17)(14,20,22,21)(15,24,23,25)(19,26,27,28)

Polycyclic presentation:

2^9|c^2=hi,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	#72308054, #72308233, #72308499
maximal normal subgroup of	#72308054, #72308233, #72308499

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961180);