#375940

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	#2404
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1190 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)(7,11)(8,12)(9,14)(10,15)(13,17)(16,27)(18,25)(19,36)(20,28)(21,29)(22,31)(23,32)(24,33)(26,35)(30,34);(5,7)(6,11)(8,19,20,35)(9,18)(10,17)(12,26,28,36)(13,15)(14,25)(16,21,33,22)(23,34)(24,29,27,31)(30,32);(5,8)(6,12)(7,16)(9,22)(10,20)(11,24)(13,27)(14,31)(15,28)(17,33)(18,35)(19,34)(21,23)(25,36)(26,30)(29,32);(1,3,4,2)(5,7)(6,13)(8,16)(9,18)(10,17)(11,15)(12,27)(14,30)(19,21)(20,33)(22,35)(23,34)(24,28)(25,32)(26,31)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,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	#72279937
maximal normal subgroup of	#72279937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283136);