#7631660

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8580 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)(5,6,9,7)(8,11,12,10)(13,14)(15,16)(17,21)(18,22)(19,20)(23,24)(25,27)(26,28);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,22)(17,25)(18,20)(21,24)(23,26)(27,28);(5,8,9,12)(6,10,7,11)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,25,28,23);(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,26)(27,28);(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,25)(26,28);(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,25)(22,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,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, 7538856);