#122393

order	512
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂²	#1459
prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

280 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,11)(4,8)(5,9,14,15)(6,10)(7,12,16,13)(17,18,20,25,22,26,29,32,21,23,19,24,27,30,28,31);(1,3,6,4)(2,7,10,16)(5,13,14,12)(8,15,11,9)(18,23)(19,20)(26,30)(28,29);(1,4,6,3)(2,8,10,11)(5,13,14,12)(7,15,16,9)(17,19,22,28,21,20,27,29)(18,24,26,31,23,25,30,32)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,d^2=h,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ci,d^b=dh,e^b=efh,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29589);