#371120

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	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

896 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)(5,6,11,17,12,18,10,16)(7,13,20,15,21,30,9,28)(8,22,27,14,25,33,26,32)(19,34,24,36,35,31,23,29);(1,3,4,2)(5,7,11,20,12,21,10,9)(6,13,17,15,18,30,16,28)(8,23,27,19,25,24,26,35)(14,31,33,29,32,34,22,36);(5,8)(6,14)(7,19)(9,24)(10,27)(11,26)(12,25)(13,29)(15,31)(16,33)(17,22)(18,32)(20,23)(21,35)(28,34)(30,36);(5,9,10,21,12,20,11,7)(6,15,16,13,18,28,17,30)(8,24,27,35,25,23,26,19)(14,31,33,29,32,34,22,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=eg,f^2=i,g^2=i,h^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	#72265222, #72307310, #72307337
maximal normal subgroup of	#72265222, #72307310, #72307337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278316);