#246994

order	512
nilpotency class	3
isoclinism	isoclinic class of #246987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#51749	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1145 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1145 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154190);