#375818

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1670 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)(7,17)(8,13)(9,22)(10,25)(11,16)(12,19)(14,23)(15,27)(18,28)(20,36)(21,29)(24,32)(26,31)(30,35)(33,34);(5,7,11,8)(6,12,16,29)(9,18,24,20)(10,21,27,19)(13,25,17,15)(14,28,31,36)(22,33,32,30)(23,35,26,34);(5,8,11,7)(6,13,16,17)(9,20,24,18)(10,19,27,21)(12,15,29,25)(14,30,31,33)(22,28,32,36)(23,34,26,35);(1,3,4,2)(5,9)(6,14)(7,18)(8,20)(10,26)(11,24)(12,28)(13,30)(15,32)(16,31)(17,33)(19,35)(21,34)(22,25)(23,27)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,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, 283014);