#7683809

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7591005);