#7739331

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×#3481	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7252 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)(7,8)(9,11)(10,12)(13,14)(15,16)(17,22)(18,19)(20,21)(23,24)(25,26)(27,28);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(14,19)(15,21)(17,25)(24,27);(6,11)(8,10)(13,15)(14,20)(16,19)(17,23)(18,21)(22,27)(24,26)(25,28);(5,8,9,10)(6,7,11,12);(5,7)(6,10)(8,11)(9,12)(13,16,18,20)(14,21,19,15)(17,24,25,27)(22,28,26,23);(13,17,18,25)(14,22,19,26)(15,23,21,28)(16,24,20,27)

Polycyclic presentation:

2^9|b^2=gh,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646527);