#7625441

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8396 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,23)(17,26)(18,21)(20,24)(22,28)(25,27);(6,9)(10,12)(13,15,18,17)(14,19,21,26)(16,22,24,25)(20,27,23,28);(13,16)(14,20)(15,22)(17,25)(18,24)(19,27)(21,23)(26,28);(16,24)(20,23)(22,25)(27,28);(13,17,18,15)(14,19,21,26)(16,25,24,22)(20,27,23,28);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,d^c=dh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532637);