#363505

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1364 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)(7,16)(8,21)(9,25)(10,14)(11,15)(12,17)(13,22)(18,32)(19,26)(20,28)(23,35)(24,29)(27,34)(30,31)(33,36);(5,7,9,17,11,19,10,20)(6,12,14,16,15,28,25,26)(8,18,22,31,23,34,24,33)(13,27,21,30,29,32,35,36);(7,17)(9,10)(12,26)(14,25)(16,28)(18,31)(19,20)(21,35)(22,24)(27,36)(30,32)(33,34);(5,8,11,23)(6,13)(7,18,19,34)(9,22,10,24)(12,27)(14,21)(15,29)(16,30)(17,31,20,33)(25,35)(26,36)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=egi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,e^c=ei,g^c=gi,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, 270701);