#7624917

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624802
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂³	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1180 conjugacy classes of subgroups; only the 536 normal subgroups are shown.

normal subgroups only

This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 536 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,26)(19,27,31,43)(20,32,34,47)(21,35,33,25)(23,36,38,29)(24,39,40,28)(30,46,45,42)(37,44,41,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(17,19,28,44,21,30,34,36,22,31,39,48,33,45,20,29)(18,23,32,42,25,37,40,43,26,38,47,46,35,41,24,27);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532113);