#373760

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1221 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)(7,15)(8,12)(9,22)(10,14)(11,19)(13,25)(16,23)(17,33)(18,29)(20,30)(21,28)(24,27)(26,35)(31,34)(32,36);(5,6)(7,16,18,11)(8,12)(9,13)(10,14)(15,19,29,23)(17,32,34,26)(20,27)(21,28)(22,25)(24,30)(31,36,33,35);(1,3,4,2)(5,7)(6,11)(8,17)(9,23)(10,18)(12,26)(13,29)(14,16)(15,22)(19,25)(20,36)(21,34)(24,35)(27,31)(28,32)(30,33);(5,8,10,21)(6,12,14,28)(7,17,18,34)(9,24,25,20)(11,26,16,32)(13,30,22,27)(15,31,29,33)(19,36,23,35)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,d^2=i,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, 280956);