#7633157

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

8500 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)(7,11)(9,12)(13,14,19,23)(15,20,18,22)(16,17,21,28)(24,25,26,27);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(15,18)(20,22)(24,26)(25,27);(5,7)(6,9)(8,11)(10,12)(13,15,19,18)(14,20,23,22)(16,24,21,26)(17,25,28,27);(13,16)(14,17)(15,24)(18,26)(19,21)(20,25)(22,27)(23,28);(13,17)(14,21)(15,25)(16,23)(18,27)(19,28)(20,26)(22,24);(13,18,19,15)(14,22,23,20)(16,26,21,24)(17,27,28,25)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540353);