#121832

order	512
nilpotency class	3
isoclinism	isoclinic class of #99154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

684 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6,9,12,8,11,15,19)(7,10,14,18,13,17,16,20)(21,22)(23,29)(24,27,32,34)(25,28)(26,31)(30,33,36,35);(1,2)(3,4)(5,7,8,13)(6,10,11,17)(9,16,15,14)(12,20,19,18)(21,23)(22,26,28,29)(24,30)(25,31)(27,33,34,35)(32,36);(2,4)(6,11)(9,15)(10,17)(14,16)(21,24)(22,27,28,34)(23,30)(25,32)(26,33,29,35)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,g^2=h,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, 29028);