#121665

order	512
nilpotency class	3
isoclinism	isoclinic class of #99154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 802 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)(7,11,12,9)(8,10)(13,14,16,21,17,20,25,28)(15,18,23,26,22,19,24,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18,20,19)(16,24)(17,22)(21,27,28,26)(23,25);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19,20,18)(16,23)(17,22)(21,27,28,26)(24,25)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ei,g^b=gh,d^c=di,e^c=ei,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28861);