#497400

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3292	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1080 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404596);