#109933

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

556 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17129);