#406802

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,29,26,31,24,30)(23,32,33,27,35,36,34,28);(1,2)(3,4)(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23)(22,27)(24,33)(25,34)(26,35)(28,31)(29,32)(30,36);(2,4)(5,7,8,11)(6,10,9,12)(21,23)(22,28)(24,33)(25,34)(26,35)(27,31)(29,36)(30,32);(14,18)(17,20)(21,24,26,25)(22,29,31,30)(23,33,35,34)(27,32,28,36)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=ef,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi

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, 313998);