#358584

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1626 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)(3,4)(5,6)(7,16,20,32)(8,21)(9,15)(10,14)(11,25)(12,18,28,19)(13,24)(17,33,34,29)(22,36,35,27)(23,31)(26,30);(2,4)(5,7,11,18,10,20,9,19)(6,12,25,16,14,28,15,32)(8,22,26,34,24,35,23,17)(13,29,31,27,21,33,30,36);(2,4)(5,8,11,26,10,24,9,23)(6,13,25,31,14,21,15,30)(7,17,18,22,20,34,19,35)(12,27,16,33,28,36,32,29);(6,14)(12,28)(13,21)(15,25)(16,32)(27,36)(29,33)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226715
maximal normal subgroup of	#72226715
maximal quotient of	#72226705

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265780);