#127811

order	512
nilpotency class	4
isoclinism	isoclinic class of #11796
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 596 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35007);