#6404189

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×#3267	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2178 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)(7,10)(8,9)(11,12)(13,14,17,16)(15,21,23,28)(18,26,22,20)(19,27,25,24);(6,9)(7,11)(13,14)(15,19)(16,17)(18,20)(21,24)(22,26)(23,25)(27,28);(13,15,22,27,17,23,18,24)(14,19,26,28,16,25,20,21);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,23,25)(18,26,22,20)(21,27,28,24);(1,3,4,2)(6,9)(10,12)(14,16)(15,23)(20,26)(24,27)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=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, 6311385);