#7154662

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2260 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)(7,11)(10,12)(13,14,18,21)(15,22,25,27)(16,20,17,26)(19,23,28,24);(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,16,23,18,25,17,24)(14,19,26,27,21,28,20,22);(2,4)(6,9)(10,12)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,28,27);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,20,21,26)(15,24,25,23)(19,27,28,22)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=i,e^2=hi,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308225, #72308294, #72308482
maximal normal subgroup of	#72308225, #72308294, #72308482

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7061858);