#363926

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2582 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)(7,15)(8,19)(9,22)(10,14)(11,18)(12,20)(13,23)(16,27)(17,31)(21,33)(24,29)(25,30)(26,34)(28,32)(35,36);(5,7)(6,11,14,25)(8,17)(9,18)(10,16)(12,24,26,31)(13,15,22,27)(19,28,33,35)(20,32)(21,29)(23,30)(34,36);(7,16)(9,23)(13,22)(15,27)(17,29)(19,33)(20,34)(28,35);(5,8,10,21)(6,12)(7,17,16,29)(9,20,23,34)(11,24)(13,19)(14,26)(15,28)(18,32,30,36)(22,33)(25,31)(27,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306371, #72306505, #72306507
maximal normal subgroup of	#72306371, #72306505, #72306507

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271122);