#6396147

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	C₂×#537	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,d^c=di,e^d=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, 6303343);