#375946

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1182 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)(5,6,10,14)(7,11,17,27)(8,18,20,29)(9,22,24,13)(12,23,28,21)(15,30,19,35)(16,32,34,26)(25,33,36,31);(5,7)(6,11)(8,19,20,15)(9,16)(10,17)(12,25,28,36)(13,26)(14,27)(18,30,29,35)(21,33,23,31)(22,32)(24,34);(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,29)(14,28)(16,33)(17,19)(18,22)(21,24)(26,35)(27,36)(30,32)(31,34);(1,3,4,2)(9,24)(13,22)(16,34)(18,29)(21,23)(26,32)(30,35)(31,33)

Polycyclic presentation:

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

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, 283142);