#362875

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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)(11,23)(15,28)(16,30)(31,33);(5,6,9,14,10,13,22,27)(7,15,18,23,17,28,21,11)(8,12,20,25,19,26,32,36)(16,29,31,34,30,35,33,24);(5,7,9,21,10,17,22,18)(6,11,14,28,13,23,27,15)(8,16,20,33,19,30,32,31)(12,24,25,35,26,34,36,29);(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,26)(14,25)(15,29)(17,30)(18,31)(21,33)(22,32)(23,34)(27,36)(28,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237224, #72239720, #72239854, …
maximal normal subgroup of	#72237224, #72239720, #72239854, …
maximal quotient of	#72237240

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270071);