#6807172

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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
prime radical	#35564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2058 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)(7,11)(8,10)(9,12)(13,14,17,21)(15,18,22,27)(16,24,26,20)(19,25,28,23);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,17,22)(14,18,21,27)(16,23,26,25)(19,20,28,24);(1,2,4,3)(13,15)(14,19)(16,25)(17,22)(18,24)(20,27)(21,28)(23,26);(6,10)(9,11)(13,15)(14,18)(16,23)(17,22)(19,24)(20,28)(21,27)(25,26);(1,3,4,2)(7,12)(9,11)(15,22)(18,27)(19,28)(23,25)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714368);