#7627178

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
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×#3542	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7410 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)(8,10)(9,12)(13,14,18,21)(15,16,23,19)(17,25,26,20)(22,27,28,24);(1,2,4,3)(5,6,8,10)(7,9,12,11)(13,14)(15,16)(17,20)(18,21)(19,23)(22,24)(25,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,27)(24,25)(26,28);(5,7)(6,9)(8,12)(10,11)(13,16,18,19)(14,15,21,23)(17,24,26,27)(20,22,25,28);(1,3,4,2)(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,27)(26,28);(13,17,18,26)(14,20,21,25)(15,22,23,28)(16,24,19,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238173, #72238199, #72238254
maximal normal subgroup of	#72238173, #72238199, #72238254
maximal quotient of	#72263772, #72263795

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534374);