#388185

order	512
nilpotency class	3
isoclinism	isoclinic class of #30437
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1075 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)(5,6,12,18)(7,19,23,33)(8,14,9,27)(10,17,11,16)(13,24,32,22)(15,26,35,29)(20,31,21,34)(25,36,28,30);(5,7)(6,13)(8,25)(9,28)(10,24)(11,22)(12,23)(14,34)(15,30)(16,33)(17,19)(18,32)(20,26)(21,29)(27,31)(35,36);(1,3,4,2)(5,8,12,9)(6,14,18,27)(7,20,23,21)(10,26,11,29)(13,30,32,36)(15,17,35,16)(19,34,33,31)(22,25,24,28);(5,9,12,8)(6,15,18,35)(7,21,23,20)(10,26,11,29)(13,31,32,34)(14,17,27,16)(19,36,33,30)(22,25,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295381);