#7627498

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×#3533	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7074 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)(7,11)(8,9)(10,12)(13,14,19,20)(15,22,17,21)(16,25,26,18)(23,24,27,28);(6,9)(10,11)(14,20)(16,26)(21,22)(23,27);(5,6)(7,10)(8,9)(11,12)(13,15,19,17)(14,21,20,22)(16,23,26,27)(18,24,25,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,18)(15,23)(17,27)(19,26)(20,25)(21,24)(22,28);(1,3,4,2)(13,17,19,15)(14,21,20,22)(16,27,26,23)(18,24,25,28);(13,18)(14,16)(15,24)(17,28)(19,25)(20,26)(21,23)(22,27)

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=i,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, 7534694);