#371118

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

894 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)(7,16,19,20)(8,21)(12,17,26,30)(13,27)(18,31,33,34)(22,24)(25,32,35,36)(28,29);(1,3,4,2)(5,6)(7,17,19,30)(8,13)(9,23)(10,14)(11,15)(12,20,26,16)(18,32,33,36)(21,27)(22,29)(24,28)(25,34,35,31);(5,7)(6,12)(8,18)(9,20)(10,16)(11,19)(13,25)(14,30)(15,26)(17,23)(21,33)(22,31)(24,34)(27,35)(28,32)(29,36);(5,8,10,22,11,21,9,24)(6,13,14,29,15,27,23,28)(7,18,16,31,19,33,20,34)(12,25,30,36,26,35,17,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306970, #72307302, #72307310
maximal normal subgroup of	#72306970, #72307302, #72307310

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278314);