#208825

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23595	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1069 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1069 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gh,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=ci,f^b=fh,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116021);