#119487

order	512
nilpotency class	3
isoclinism	isoclinic class of #99154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1382 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^b=gh,e^c=eh,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, 26683);