#116367

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
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

822 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=eh

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, 23563);