#6393408

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3541	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,23,19,24,17,22)(15,20,28,36,27,35,26,34)(16,29,30,21,32,37,31,38)(25,39,40,33,41,43,42,44);(5,7)(6,9)(8,12)(10,11)(13,15,19,27)(14,20,24,35)(16,25,32,41)(17,26,18,28)(21,33,38,44)(22,34,23,36)(29,39,37,43)(30,40,31,42);(1,2,3,4)(13,16)(14,21)(15,25)(17,30)(18,31)(19,32)(20,33)(22,37)(23,29)(24,38)(26,40)(27,41)(28,42)(34,43)(35,44)(36,39);(7,12)(9,11)(13,17,19,18)(14,22,24,23)(15,26,27,28)(16,30,32,31)(20,34,35,36)(21,37,38,29)(25,40,41,42)(33,43,44,39);(6,10)(9,11)(15,27)(20,35)(25,41)(26,28)(33,44)(34,36)(39,43)(40,42)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^b=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6300604);