#331632

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

560 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egi,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

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, 238828);