#7674778

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	#37678	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6938 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581974);