#508861

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33171	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343806, #72343820, #72343829
maximal normal subgroup of	#72343806, #72343820, #72343829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416057);