#215763

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(30,31)(33,34,36,39,37,40,43,46)(35,38,42,45,41,44,47,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,23)(24,31)(25,26)(27,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,29)(18,24,23,31)(20,28,22,30)(25,27,26,32)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(2,8)(6,9)(7,16)(14,15)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)(35,41)(38,44)(42,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122959);