#7695614

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	#37683	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7602810);