#495349

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

824 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=h,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,e^b=eh,g^b=ghi,h^b=hi,d^c=di

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, 402545);