#368084

order	512
nilpotency class	3
isoclinism	isoclinic class of #23882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1621 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)(5,6,11,16)(7,12,9,14)(8,13,21,29)(10,15,24,32)(17,28,20,25)(18,26,23,31)(19,34,35,27)(22,30,33,36);(1,3,4,2)(5,7,11,9)(6,12,16,14)(8,19,21,35)(10,23,24,18)(13,27,29,34)(15,31,32,26)(17,33,20,22)(25,36,28,30);(5,8,10,22,11,21,24,33)(6,13,15,30,16,29,32,36)(7,17,18,19,9,20,23,35)(12,25,26,27,14,28,31,34);(5,9,11,7)(6,14,16,12)(8,20,21,17)(10,23,24,18)(13,28,29,25)(15,31,32,26)(19,22,35,33)(27,30,34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,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, 275280);