#7627580

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×#3549	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7640 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)(7,8)(9,10)(11,12)(13,14,19,20)(15,24,17,21)(16,18,23,22)(25,28,27,26);(6,10)(7,11)(14,20)(16,23)(21,24)(25,27);(5,7,9,11)(6,8,10,12)(13,15,19,17)(14,21,20,24)(16,25,23,27)(18,26,22,28);(5,8)(6,11)(7,10)(9,12)(13,16,19,23)(14,22,20,18)(15,25,17,27)(21,28,24,26);(1,3,4,2)(13,17,19,15)(14,21,20,24)(16,27,23,25)(18,26,22,28);(13,18,19,22)(14,23,20,16)(15,26,17,28)(21,27,24,25)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=i,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, 7534776);