#416201

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1316 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319999, #72320048, #72320063, …
maximal normal subgroup of	#72319999, #72320048, #72320063, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323397);