#363927

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

2566 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)(7,12)(8,19)(9,11)(10,15)(13,20)(14,23)(16,27)(17,31)(18,29)(21,25)(22,24)(26,32)(28,35)(30,33)(34,36);(6,11,15,14)(7,16)(12,24,27,29)(13,25,28,19)(17,30)(18,22)(26,31,36,33)(32,34);(5,7,10,16)(6,12,15,27)(8,17,21,30)(9,22,23,18)(11,24,14,29)(13,26,28,36)(19,33,25,31)(20,34,35,32);(5,8,10,21)(6,13)(7,17,16,30)(9,20,23,35)(11,25)(12,26)(14,19)(15,28)(18,32,22,34)(24,31)(27,36)(29,33)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,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	#72306347, #72306505, #72306516
maximal normal subgroup of	#72306347, #72306505, #72306516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271123);