#6467696

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250360, #72256847, #72256945, …
maximal normal subgroup of	#72250360, #72256847, #72256945, …
maximal quotient of	#72264372, #72264519, #72265088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374892);