#6372718

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4560 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341746, #72341788
maximal normal subgroup of	#72341746, #72341788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279914);