#204337

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×C₄×D₄
prime radical	C₂×#466	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286991, #72287017, #72287041, …
maximal normal subgroup of	#72286991, #72287017, #72287041, …
maximal quotient of	#72287494, #72323083

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111533);