#7654579

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
prime radical	#37614	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561775);