#6399982

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,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, 6307178);