#368225

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

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

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, 275421);