#267808

order	512
nilpotency class	3
isoclinism	isoclinic class of #267755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175004);