#6389161

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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×#3055	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2188 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)(13,14)(15,18)(16,24)(17,21)(19,23)(20,27)(22,25)(26,28);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15,18,14)(16,22,26,20)(17,21,19,23)(24,27,28,25);(13,16,19,27,18,26,17,25)(14,20,21,24,15,22,23,28);(7,12)(9,11)(13,17,18,19)(14,21,15,23)(16,25,26,27)(20,24,22,28);(6,10)(9,11)(14,15)(16,26)(21,23)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296357);