#195985

order	512
nilpotency class	3
isoclinism	isoclinic class of #195981
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1098 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6,8,12,9,11,15,18)(7,10,14,17,13,16,19,20)(21,22)(23,27,33,40)(24,28)(25,29)(26,30)(31,38,46,50)(32,39,48,52)(34,41)(35,42)(36,43)(37,44)(45,49,47,51);(5,7,8,14,9,13,15,19)(6,10,12,17,11,16,18,20)(21,23,26,33)(22,27)(24,31,35,46)(25,32,37,48)(28,38)(29,39)(30,40)(34,45,36,47)(41,49)(42,50)(43,51)(44,52);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,26,35)(22,28,30,42)(23,31,33,46)(25,36,37,34)(27,38,40,50)(29,43,44,41)(32,47,48,45)(39,51,52,49);(6,11)(10,16)(12,18)(17,20)(21,25,26,37)(22,29,30,44)(23,32,33,48)(24,34,35,36)(27,39,40,52)(28,41,42,43)(31,45,46,47)(38,49,50,51)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,e^b=eh,g^b=gh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103181);