#410679

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322498
maximal normal subgroup of	#72322498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317875);