#368243

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	#2404
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1480 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)(5,6,10,13)(7,11,16,26)(8,18)(9,14,23,29)(12,20)(15,24)(17,25,32,36)(19,27)(21,30)(22,28)(31,35)(33,34);(1,3,4,2)(5,7,10,16)(6,11,13,26)(8,19)(9,17,23,32)(12,27)(14,25,29,36)(15,22)(18,34)(20,33)(21,31)(24,30)(28,35);(5,8,9,22,10,20,23,21)(6,12,14,30,13,18,29,28)(7,15,17,33,16,31,32,19)(11,24,25,34,26,35,36,27);(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,26)(14,25)(18,35)(19,21)(20,31)(22,33)(23,32)(27,28)(29,36)(30,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280196, #72318268, #72318503, …
maximal normal subgroup of	#72280196, #72318268, #72318503, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275439);