#7627530

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×#3558	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7370 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)(5,6,9,7)(8,11,12,10)(13,14,18,19)(15,22,26,20)(16,17,23,21)(24,27,28,25);(6,7)(10,11)(14,19)(16,23)(20,22)(24,28);(5,7,9,6)(8,11,12,10)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,27)(23,28);(5,8,9,12)(6,10,7,11)(13,16,18,23)(14,21,19,17)(15,24,26,28)(20,27,22,25);(1,3,4,2)(13,15)(14,22)(16,24)(17,27)(18,26)(19,20)(21,25)(23,28);(13,17,18,21)(14,23,19,16)(15,25,26,27)(20,28,22,24)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=gi,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, 7534726);