#7626641

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3480	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533837);