#373869

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1268 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)(7,19)(8,24)(9,27)(10,17)(11,16)(12,18)(13,22)(14,26)(15,28)(20,31)(21,36)(23,35)(25,32)(29,34)(30,33);(5,7,10,22,12,23,11,25)(6,13,16,19,18,32,17,35)(8,20,26,9,21,28,30,29)(14,15,24,27,33,31,36,34);(1,3,4,2)(5,8)(6,14)(7,20)(9,22)(10,26)(11,30)(12,21)(13,31)(15,32)(16,24)(17,36)(18,33)(19,34)(23,28)(25,29)(27,35);(5,9,10,28,12,29,11,20)(6,15,16,27,18,31,17,34)(7,21,22,30,23,8,25,26)(13,24,19,33,32,36,35,14)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214895, #72265334, #72265359, …
maximal normal subgroup of	#72214895, #72265334, #72265359, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281065);