#189511

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320225, #72320236, #72320245
maximal normal subgroup of	#72320225, #72320236, #72320245
maximal quotient of	#72320759

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96707);