#238072

order	512
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1640 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1640 conjugacy classes of subgroups — too many to draw. Only its 178 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,22,25,16)(13,17)(14,18)(19,32,23,27)(20,28)(21,29)(24,35,37,30)(26,31)(33,40,36,38)(34,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,23,25,19)(13,20)(14,21)(16,27,22,32)(17,28)(18,29)(24,36,37,33)(26,34)(30,38,35,40)(31,39);(5,8)(9,12)(10,16)(11,19)(13,24)(14,25)(15,27)(17,30)(18,22)(20,33)(21,23)(26,37)(28,38)(29,32)(31,35)(34,36)(39,40);(9,13,14,26)(10,17,18,31)(11,20,21,34)(12,24,25,37)(15,28,29,39)(16,30,22,35)(19,33,23,36)(27,38,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303835
maximal normal subgroup of	#72303835
maximal quotient of	#72303962

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145268);