#7500883

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500740
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fi,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318894, #72318925, #72318957
maximal normal subgroup of	#72318894, #72318925, #72318957

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408079);