#6584087

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2130 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282046, #72282899
maximal normal subgroup of	#72282046, #72282899

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491283);