#7712328

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×#3549	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7376 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(14,18)(19,21)(20,24)(26,27);(6,8)(10,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(5,8,9,6)(7,11,12,10);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7619524);