#121725

order	512
nilpotency class	3
isoclinism	isoclinic class of #99134
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

830 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=hi,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^b=gi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28921);