#7681808

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×#3500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8032 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)(6,9)(10,11)(13,14,19,16)(15,20,22,17)(18,24,26,21)(23,25,28,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,17)(18,23)(19,22)(21,27)(24,25)(26,28);(5,6)(7,11)(8,9)(10,12)(15,22)(17,20)(23,28)(25,27);(5,7,8,12)(6,10,9,11)(13,16,19,14)(15,17,22,20)(18,24,26,21)(23,25,28,27);(13,17)(14,15)(16,22)(18,25)(19,20)(21,23)(24,28)(26,27);(13,18,19,26)(14,21,16,24)(15,23,22,28)(17,25,20,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=hi,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, 7589004);