#6403287

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×#3334	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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)(15,21)(17,22)(19,26)(20,25)(23,27)(24,28);(5,7)(6,9)(8,12)(10,11)(13,14,18,16)(15,19,24,23)(17,20,22,25)(21,26,28,27);(13,15,22,28,18,24,17,21)(14,19,25,27,16,23,20,26);(7,12)(9,11)(13,16,18,14)(15,23,24,19)(17,25,22,20)(21,27,28,26);(1,3,4,2)(6,10)(9,11)(14,16)(19,23)(20,25)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,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, 6310483);