#368398

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2408
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321498, #72321530, #72321603, …
maximal normal subgroup of	#72321498, #72321530, #72321603, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275594);