#129740

order	512
nilpotency class	4
isoclinism	isoclinic class of #124657
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

958 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^d=fi

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, 36936);