#191291

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#383
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

522 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

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

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, 98487);