#232191

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

500 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efh,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,e^a=eh,c^b=cefi,d^b=de,f^b=fh,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139387);