#8072037

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6442 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7979233);