#6380175

order	512
nilpotency class	3
isoclinism	isoclinic class of #61106
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2482 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)(6,12)(7,16)(9,10)(13,27)(14,26)(15,23)(17,30)(18,19)(20,29)(21,24)(28,36)(32,34);(5,6)(7,13)(8,20)(9,23)(10,12)(11,15)(14,21)(16,25)(17,31)(18,33)(19,27)(22,26)(24,29)(28,30)(32,35)(34,36);(5,7,9,18,11,19,10,16)(6,13,23,33,15,27,12,25)(8,17,24,34,22,32,21,30)(14,28,20,31,29,36,26,35);(5,8)(6,14)(7,17)(9,24)(10,21)(11,22)(12,26)(13,28)(15,29)(16,30)(18,34)(19,32)(20,23)(25,35)(27,36)(31,33);(6,15)(12,23)(13,27)(14,29)(20,26)(25,33)(28,36)(31,35)

Polycyclic presentation:

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

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, 6287371);