#6396725

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fgh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,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, 6303921);