#7627434

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3540	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7112 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534630);