#362663

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

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, 269859);