#7627308

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×#3485	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8412 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)(13,14,19,24)(15,18,23,20)(16,22,17,21)(25,28,26,27);(6,7)(10,11)(13,15)(14,20)(16,25)(17,26)(18,24)(19,23)(21,28)(22,27);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,16,19,17)(14,21,24,22)(15,25,23,26)(18,27,20,28);(5,8)(6,10)(7,11)(9,12)(13,17,19,16)(14,22,24,21)(15,26,23,25)(18,28,20,27);(13,15)(14,18)(16,25)(17,26)(19,23)(20,24)(21,27)(22,28);(13,18)(14,23)(15,24)(16,27)(17,28)(19,20)(21,26)(22,25)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534504);