#232808

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7,4)(3,11,15,12)(5,10,6,9)(8,14,16,13)(17,18,23,27)(19,28,32,40)(20,33,37,43)(21,38,22,26)(24,30,41,31)(25,36,44,35)(29,46,47,39)(34,45,48,42);(1,3,7,15)(2,8,4,16)(5,14,6,13)(9,12,10,11)(17,19,23,32)(18,24,27,41)(20,34,37,48)(21,30,22,31)(25,42,44,45)(26,40,38,28)(29,35,47,36)(33,39,43,46);(1,4,7,2)(3,12,15,11)(5,9,6,10)(8,13,16,14)(17,20,23,37)(18,25,27,44)(19,29,32,47)(21,36,22,35)(24,39,41,46)(26,43,38,33)(28,45,40,42)(30,48,31,34);(1,5,7,6)(2,9,4,10)(3,13,15,14)(8,11,16,12)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,37,36)(24,40,41,28)(25,33,44,43)(29,34,47,48)(39,45,46,42)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140004);