#392325

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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)(7,15,18,11)(8,19,20,34)(9,14)(10,13)(12,21,25,35)(16,29)(17,28,32,24)(22,26)(23,31)(27,36)(30,33);(5,7,9,17,10,18,22,32)(6,11,14,24,13,15,26,28)(8,16,21,31,20,30,35,36)(12,23,19,33,25,27,34,29);(5,8)(6,12,13,25)(7,16)(9,21)(10,20)(11,23,15,27)(14,19,26,34)(17,31)(18,30)(22,35)(24,33,28,29)(32,36);(2,4)(8,20)(12,25)(16,30)(19,34)(21,35)(23,27)(29,33)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,d^c=di,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299521);