#375185

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,g^c=gi

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, 282381);