#6389137

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2114 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296333);