#7968098

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	#34681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7875294);