#365078

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	#2535
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,h^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,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249991, #72250004, #72250014, …
maximal normal subgroup of	#72249991, #72250004, #72250014, …
maximal quotient of	#72264953

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272274);