#368418

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
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

1324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250079, #72250080, #72250092, …
maximal normal subgroup of	#72250079, #72250080, #72250092, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275614);