#195156

order	512
nilpotency class	3
isoclinism	isoclinic class of #18224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102352);