#374762

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281958);