#395221

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1819 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:

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

Polycyclic presentation:

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

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, 302417);