#116879

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
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	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

770 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 146 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)(6,10)(7,14)(8,11)(13,19)(16,17)(18,20)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(5,6,11,19,9,13,8,10)(7,15,14,18,17,12,16,20)(21,23)(22,26)(24,29)(25,30)(27,34)(28,33)(31,36)(32,35);(5,7)(6,12)(8,16)(9,17)(10,18)(11,14)(13,15)(19,20)(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=dh,c^2=i,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=eh

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, 24075);