#6374689

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

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×#3494	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,g^c=gi,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, 6281885);