#376261

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2408
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1312 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)(5,6,11,16)(7,17,19,29)(8,12,21,13)(9,22,25,36)(10,26,23,15)(14,30,31,24)(18,28,33,27)(20,32,34,35);(5,7,10,8,11,19,23,21)(6,12,15,17,16,13,26,29)(9,18,24,20,25,33,30,34)(14,27,22,32,31,28,36,35);(5,8,10,19,11,21,23,7)(6,13,15,29,16,12,26,17)(9,20,24,33,25,34,30,18)(14,28,22,35,31,27,36,32);(1,3,4,2)(5,9,23,30,11,25,10,24)(6,14,26,36,16,31,15,22)(7,18,21,34,19,33,8,20)(12,27,29,35,13,28,17,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214895, #72312195
maximal normal subgroup of	#72214895, #72312195

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283457);