#183229

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1174 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)(3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,18)(16,17)(19,20)(21,22,25,29,26,24,28,33)(23,27,32,35,31,30,34,36);(2,4)(5,7,9,17,10,19,8,18)(6,11,12,20,14,16,13,15)(21,23)(22,27)(24,30)(25,32)(26,31)(28,34)(29,35)(33,36);(5,6)(7,16)(8,13)(9,12)(10,14)(11,19)(15,17)(18,20)(21,24,25,33,26,22,28,29)(23,30,32,36,31,27,34,35);(5,8,10,9)(6,12,14,13)(7,17,19,18)(11,15,16,20)(22,24)(23,31)(29,33)(32,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,d^a=di,e^a=eh,g^a=gh,c^b=ch,d^b=dhi,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319210, #72319531, #72319546
maximal normal subgroup of	#72319210, #72319531, #72319546
maximal quotient of	#72323122

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90425);