#7681049

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	C₂×#3489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7965 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)(3,4)(6,9)(10,11)(13,14,19,16)(15,23,24,20)(17,21,27,26)(18,25,28,22);(5,6)(7,10)(8,9)(11,12)(13,15,19,24)(14,20,16,23)(17,22,27,25)(18,21,28,26);(2,4)(5,6)(7,11)(8,9)(10,12)(15,24)(17,27)(20,23)(21,26);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,16,19,14)(15,23,24,20)(17,26,27,21)(18,25,28,22);(13,17)(14,21)(15,25)(16,26)(18,20)(19,27)(22,24)(23,28);(13,18,19,28)(14,22,16,25)(15,21,24,26)(17,20,27,23)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,f^2=i,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, 7588245);