#409958

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1768 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1768 conjugacy classes of subgroups — too many to draw. Only its 258 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)(7,9)(8,10)(11,12)(14,19)(16,17)(20,27)(21,24)(22,25)(26,28);(2,4)(5,7,8,11)(6,9,10,12)(13,14,18,21)(15,20,23,28)(16,19,17,24)(22,26,25,27);(1,2,3,4)(13,15)(14,20)(16,22)(17,25)(18,23)(19,26)(21,28)(24,27);(6,10)(9,12)(13,16,18,17)(14,19,21,24)(15,22,23,25)(20,26,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317154);