#375760

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1683 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)(7,18)(8,14)(9,21)(10,28)(11,16)(12,17)(13,20)(15,27)(19,30)(22,23)(24,34)(25,32)(26,33)(29,36)(31,35);(5,7)(6,13)(8,22)(9,17)(10,21)(11,20)(12,15)(14,30)(16,27)(18,28)(19,25)(23,34)(24,31)(26,36)(29,32)(33,35);(5,8,12,26)(6,14,17,33)(7,19,15,31)(9,23,13,29)(10,25,11,24)(16,34,28,32)(18,35,27,30)(20,22,21,36);(1,3,4,2)(5,9)(6,15)(7,17)(8,23)(10,18)(11,27)(12,13)(14,31)(16,20)(19,33)(21,28)(22,34)(24,30)(25,35)(26,29)(32,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,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, 282956);