#368899

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276095);