#6374686

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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×#3460	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3170 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)(23,33)(24,28)(25,31)(26,32)(27,30)(29,35)(34,36);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23)(22,28)(24,26)(25,35)(27,34)(29,30)(31,36)(32,33);(7,14)(10,13)(15,20)(17,19)(21,24,26,23)(22,29,32,36)(25,35,27,34)(28,31,33,30);(6,11)(10,13)(12,18)(17,19)(21,25,26,27)(22,30,32,31)(23,34,24,35)(28,36,33,29);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(21,25,26,27)(22,31,32,30)(23,34,24,35)(28,29,33,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,e^b=ei,g^c=gi

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, 6281882);