#364104

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,19)(18,26)(20,32)(21,27)(22,29)(23,30)(24,31)(25,33)(28,36)(34,35);(5,7,11,20,10,19,24,9)(6,12,16,14,15,17,31,32)(8,21,22,35,23,36,25,18)(13,27,29,34,30,28,33,26);(5,8)(6,13)(7,18)(9,21)(10,23)(11,25)(12,26)(14,28)(15,30)(16,33)(17,34)(19,35)(20,36)(22,24)(27,32)(29,31);(5,9,24,19,10,20,11,7)(6,14,31,12,15,32,16,17)(8,21,22,35,23,36,25,18)(13,28,29,26,30,27,33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265105, #72265644, #72265655, …
maximal normal subgroup of	#72265105, #72265644, #72265655, …
maximal quotient of	#72265657, #72265662, #72286739

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271300);