#454466

order	512
nilpotency class	3
isoclinism	isoclinic class of #453952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ei,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,d^c=de,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361662);