#409953

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1713 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1713 conjugacy classes of subgroups — too many to draw. Only its 274 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239029, #72239132, #72239138
maximal normal subgroup of	#72239029, #72239132, #72239138

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317149);