#7627553

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×#3489	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7990 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=h,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, 7534749);