#6403919

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
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×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 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)(15,20)(16,26)(17,22)(18,23)(19,25)(21,28)(24,27);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,25,16)(17,18,21,27)(22,23,28,24);(6,10)(9,11)(13,16,27,22,19,20,18,28)(14,21,23,15,25,17,24,26);(7,12)(9,11)(13,17)(14,22)(15,18)(16,24)(19,21)(20,23)(25,28)(26,27);(1,3,4,2)(6,10)(9,11)(13,18,19,27)(14,23,25,24)(15,17,26,21)(16,28,20,22)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311115);