#331018

order	512
nilpotency class	3
isoclinism	isoclinic class of #331007
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

469 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 124 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=de,e^c=eh,f^c=fh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238214);