#6607031

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3282	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2306 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)(7,10)(8,11)(9,12)(13,14,19,24)(15,25,28,35)(16,30,33,37)(17,22,18,23)(20,29,36,27)(21,31,38,32)(26,34,41,44)(39,40,43,42);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,27,19,28,18,29)(14,20,23,25,24,36,22,35)(16,26,31,40,33,41,32,42)(21,34,37,39,38,44,30,43);(7,12)(9,10)(13,16,18,32,19,33,17,31)(14,21,22,30,24,38,23,37)(15,26,29,42,28,41,27,40)(20,34,35,43,36,44,25,39);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,31,33,32)(20,35,36,25)(21,30,38,37)(26,40,41,42)(34,43,44,39);(7,12)(9,10)(15,28)(20,36)(25,35)(26,41)(27,29)(34,44)(39,43)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=g,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514227);